Submitted to GRF2006 Symposium Proceedings, Springer Verlag IAG Series, 12/2006, Revised 1/2008
Seasonal Position Variations
Reference Frame Realization
and
1
Regional
J. T. Freymueller
Geophysical Institute,
University of Alaska Fairbanks, 903 Koyukuk Drive, Fairbanks, AK 99775 USA
Abstract. Many GPS sites show significant seasonal position variations (especially in the vertical), resulting from a combination of geophysical
loading and systematic errors. It is common to
realize a regional reference frame or align a time
series with a model by transforming each epoch
solution into ITRF (or a local or regional frame)
using a regional set of sites, where these reference
frame sites are assumed to move linearly with
time. Seasonal variations in the frame sites can
bias this frame realization and the seasonal variations are then aliased into the site coordinates.
I estimate seasonal variations at a set of sites
across northern North America and the Arctic using an iterative non-parameteric approach. Seasonal variations at most sites exceed 4-5 mm in the
vertical. A test with synthetic time series shows
that ignoring these variations when realizing a
daily or weekly reference frame can cause apparent seasonal variations in other sites as large as 4
mm amplitude. We should work toward a linear +
seasonal model as part of our reference frame
definition.
Keywords. Reference frame, GPS, seasonal displacement.
1 Introduction
Seasonal variations in positions are commonly
observed, especially in height component. For
example, seasonal variations in height for sites in
Japan are significant, and spatially coherent in
phase and amplitude (Murakami and Miyazaki,
2001). Heki (2001, 2003) showed that these variations could be explained primarily in terms of hydrological loading, mainly due to snow loading in
northern Honshu. Dong et al. (2002) analyzed
contributions to seasonal variations globally.
Hydrological loading can cause substantial
variations in vertical displacement, and these
variations are often annual in period. For example,
van Dam et al. (2001) calculated model vertical
displacements from the convolution of loads predicted by a global hydrological model with elastic
surface loading Greens functions (Farrell, 1972).
The resulting model displacements had peak-to-
peak extreme values (over several years) of 10-30
mm in many places, and a predominantly annual
period, and were in-phase with the observed seasonal variations at many sites. They further found
that the variance reduction in height variations at
147 globally distributed GPS sites when the model
variations were removed was consistent with that
expected from the variance of the model. However, as noted by Ray (2006), systematic errors in
the GPS system and models can also produce seasonal variations, so matching loading models to
data in detail has proven difficult.
Fig. 1 Sites used in this study. White triangles show
sites in the solutions used in the study, and black triangles are sites used to realize reference frame. No sites in
Alaska are used to realize the reference frame because
of earthquake displacements and non-linear postseismic
motion, common problems in many regional networks.
The gray triangles show the location of two test sites
discussed in section 5.
It is less widely recognized that seasonal variations, if present in the data, will bias the reference
frame transformation parameters estimated as part
of regional reference frame realization, and the
alignment of epoch solutions with a long-term
velocity model. In the case of small networks, this
is exploited to use the regional frame as a regional
filter (e.g., Wdowinski et al., 1997). In the case of
global networks, only long-wavelength variations,
such as low degree global loading (e.g., Blewitt et
al., 2001), may remain to bias frame parameters.
However, many regional networks are of intermediate scale, and as I will show, seasonal variations
in positions of sites used to realize the frame can
Submitted to GRF2006 Symposium Proceedings, Springer Verlag IAG Series, 12/2006, Revised 1/2008
have a substantial impact on the effective reference frame and the resulting position time series.
2 Data Set
I study the impact of seasonal variations on reference frame realization using 6 years of daily
solutions from a continuous GPS network spanning the northern part of North America and surrounding regions (Figure 1). The network is focused on studying active tectonic and volcanic
deformation in Alaska. The number of sites used
in the solutions varies from about 60 to more than
120 over the time span studied, with the number of
sites increasing steadily with time.
Each daily solution was analyzed as a network
solution using the GIPSY-OASIS software version
GOA4. A uniform data analysis strategy was used
for all data. Every day, data from all stations were
used simultaneously to estimate satellite clocks,
station positions, time-dependent tropospheric
path delays and gradients, and phase ambiguities. I
used an ambiguity-free solution for this analysis.
One station, usually ALGO, had its clock held
fixed as a reference clock. Orbits were fixed to the
JPL non-fiducial orbit, a frame-free orbit based on
JPL’s IGS submission, and the JPL-estimated
EOPs were also used here. I applied the IGS relative antenna phase center models without radome
corrections, and ocean tidal loading corrections
based on the TPXO.2 tidal model. Outliers were
removed from the time series; most outliers resulted from incomplete days of data.
Each daily solution is in an indeterminate
frame, and must be transformed into ITRF by estimating a 7-parameter transformation between the
daily coordinate estimates and the ITRF model
prediction for that day. I evaluate ITRF2000
(IGSb00) at each day, and use 17 sites to estimate
the transformation (solid triangles in Figure 1).
The sites are weighted according to the combined
uncertainty in the ITRF predicted position and the
daily estimate for that site. The reference frame
sites were chosen because they have welldetermined velocities in ITRF, and do not have
any known problems or anomalies in their time
series that might bias the frame estimation. A few
of the sites do have small discontinuities identified
within this time period. I compared daily position
estimates where the frame was realized both with
and without these sites, and the effect of the discontinuities on daily position estimates is at no
more than the mm level.
3 Non-parametric Estimation of Seasonal Variation
2
Sinusoidal parametric functions are commonly
used to estimate seasonal components in geodetic
time series. However, some potential seasonal
loading sources will not show a perfectly sinusoidal form, and a non-parametric approach may be
more suitable. For example, snow loading should
vary in amplitude with the weight of the snow
load, which in high northern latitudes increases
from October or November through March, and
then rapidly melts in the spring. I use a nonparametric approach to estimate seasonal deformation to avoid any assumptions about the source
other than an annual period. A parametric approach using a sufficient number of sinusoids
should produce equivalent results, but the nonparametric approach also allows for easy temporal
smoothing of the estimated seasonal variation.
Fig. 2 Original (top) and corrected (bottom) detrended
time series for GUS2. Blue dots are daily height estimates, and red dots are weekly averages. The black
curve on the top figure is the estimated seasonal variation (Figure 3). Vertical axis spans +/- 30 mm.
The method is similar to techniques used in a
variety of other applications, including the removal of seasonal variations from tide gauge data
(e.g., Larsen et al., 2003). The time series is detrended (Figure 2, top), and the residuals are sorted
into seasonal bins by fractional year. I used 40
bins to represent the full year, so each bin represents 8-9 days. The first bin consists of the first
days of the year, January 1-9, and the average residual for January 1-9 of all years is the mean seasonal correction for that bin (Figure 3). I then applied a 5-point smoother to the raw bin averages to
get a smoothed seasonal correction. The seasonal
averages were then interpolated and subtracted
from the original time series to get a corrected
time series (Figure 2, bottom).
Figures 2 and 3 show the time series for a site in
Submitted to GRF2006 Symposium Proceedings, Springer Verlag IAG Series, 12/2006, Revised 1/2008
Gustavus, Alaska (GUS2). The station height
reaches its lowest point on average at about 3035% of the way through the year, or in April. This
is the time of year when significant melt and runoff begins, as temperatures rise above the melting
on a consistent basis. The site reaches its highest
point between 70% and 100% of the way through
the year, or September to December. The timing of
the start of significant snow accumulation varies
from year to year, but usually falls within this time
range. Interannual variations are also clearly visible to the eye in this time series. The timing of this
record suggests that snow loading dominates the
seasonal variation at this site, and other highlatitude sites show similar timing.
3
through an iterative approach. To estimate the
seasonal variation at one frame site, I first realize
the frame using the other N-1 sites, and I use the
procedure outlined in section 3 to estimate the
seasonal contribution. This process is repeated for
all sites. I then repeat the entire procedure, this
time using a model consisting of linear ITRF velocities plus the seasonal correction from the previous iteration. For most sites, this process converged within 2-3 iterations. For the sites YELL
and IRKT (and to a lesser extent, TIXI), however,
convergence was not reached until 4-5 iterations,
because both sites have strong seasonal variations
and the seasonal corrections converged in a
damped oscillatory fashion toward convergence.
Fig. 3 Estimated seasonal variation for GUS2. Blue dots
are daily height estimates, plotted by fractional year.
Black curve is interpolated from the raw bin averages,
and red curve on the smoothed bin averages. Vertical
axis spans +/- 30 mm. The figure shows 1.5 years of
data to emphasize the seasonality of the residuals.
4 Seasonal Variations at Frame Sites
The sites used to realize the reference frame
also are expected to have seasonal variations. Because we normally ignoring seasonal variations
and realize the reference frame using a linear
model in time for the station coordinates, the seasonal variations will bias the estimated frame parameters and, through the frame, may be aliased
into the positions of all sites. The process of realizing the frame may “damp down” actual seasonal
variations at the sites used to define the frame for
the same reason: some component of the seasonal
variation in the sites will leak into the frame estimate. Therefore, seasonal variations at these sites
should be estimated using time series that are generated without using that site to define the frame.
I evaluate the seasonal variations at frame sites
Fig. 4 Estimated seasonal variations at two sites: FLIN
(top) and YELL (bottom). Black curve is seasonal
height estimate, red curve is seasonal north estimate,
and blue curve is seasonal east estimate. Vertical axis
spans +/- 10 mm, horizontal axis in years.
At most sites, the seasonal variation was the
largest in the vertical component, as would be expected if loading was the major cause of the seasonal variation (Figure 4). Only four sites had seasonal height variations smaller than 4 mm ampli-
Submitted to GRF2006 Symposium Proceedings, Springer Verlag IAG Series, 12/2006, Revised 1/2008
tude: DUBO, FLIN, STJO and TIXI. Five sites
showed vertical seasonal variations of ~4-5 mm
amplitude: ALGO, CHUR, MDO1, NLIB, DRAO.
The remaining sites had vertical seasonal amplitudes in excess of 5 mm, although none were
greater than 10 mm amplitude.
Few of the estimated seasonal patterns were sinusoidal. At most sites, the high and low parts of
the seasonal curve were asymmetrical, and in most
cases the low part of the curve covers less than
half the year (Figure 4 bottom, site YELL). In general, the sites begin to uplift between 1/3 and 1/2
of the way through the year, and begin to subside
late in the year. Qualitatively, this matches the
pattern at GUS2, although not all sites are in the
sub-arctic and each site’s pattern is different.
4
at this level by non-geophysical noise. The same
holds for the adjacent sites NYAL and NYA1 in
Ny Ålesund, Norway.
The fact that several iterations were needed for
convergence shows that time series can change
significantly when seasonal variations are included
in the reference frame definition. In the case of
Irkutsk (Figure 5), this change can be significant
and easily visible. Irkutsk is located at the edge of
the network, and far from any site other than TIXI.
Any bias in realizing the reference frame would be
amplified in the position of this remote site, thanks
to the long lever arm between IRKT and the centroid of the network. The final estimated time series for IRKT shows a smaller seasonal variation
than the initial time series based on the unmodified
ITRF, which shows that the additional variations
in the time series result from aliasing of reference
frame realization errors into the site coordinates.
This change is particularly notable in the first half
of the time series, where the variation is dominated by an annual period. After the summer of
2003, the time series is dominated by shorted period variations. The cause of this change is not
certain, but the pattern is unique to this site.
5 Impacts on Time Series
Fig. 5 Time series for Irkutsk (IRKT) where the reference frame is realized via an estimated transformation
(top) with the standard linear model for ITRF, (middle)
with the linear model plus seasonal corrections. The
bottom panel shows the difference between the two
series. In both cases, IRKT is not used in the alignment
of the daily solution with the frame. When seasonal
variations at the other sites are accounted for, the seasonal variation at IRKT is visibly reduced.
There are two co-located sites at Tromsø, Norway, TROM and TRO1. The two sites have different monuments, with TROM located on a tower
and TRO1 on a pillar atop a building. The vertical
seasonal variation is almost identical at these two
sites, ~7 mm amplitude, suggesting that it is dominated by a geophysical signal and not simply error
in the time series. Ray (2006) showed that electrical effects such as multipath and scattering can
cause significant apparent seasonal variations, but
with different local environments this error source
should be different at the two sites. However, the
horizontal seasonal variations at the two sites are
quite different, with the estimated variation curves
differing by 2-3 mm during much of the year, so it
is possible that the seasonal variations are biased
I further evaluated the impact of seasonal variations in the frame sites on time series using a test
with synthetic data. If seasonal variations are present in frame sites, but ignored, how much error
will be aliased into apparent seasonal biases in
other sites? I generated a one-year daily synthetic
time series for all of the frame sites based on their
ITRF positions and velocities and the final estimated seasonal variation. I also generated synthetic time series for two test sites, which had no
seasonal variations. The test sites were located in
Fairbanks, Alaska, and on Attu Island, western
Aleutians, to represent a typical point in the center
of the area of main interest, and a site near the
edge of the network (Figure 1). Each daily synthetic solution was transformed into ITRF in the
same way as the actual data, ignoring the seasonal
variation at the frame sites.
Seasonal variations were found in the resulting
time series of test sites, which indicates that aliasing of seasonal variation at the frame sites into the
reference frame parameters, and then into all site
coordinates, biased the ultimate site coordinates.
Both test sites showed apparent seasonal variations, but of different magnitude. The error from
ignoring seasonal variations at the frame sites thus
varies with space. Figure 6 shows the results for
the test site in the western Aleutians. The time
Submitted to GRF2006 Symposium Proceedings, Springer Verlag IAG Series, 12/2006, Revised 1/2008
series shows apparent horizontal seasonal variations of 1-2 mm amplitude, and vertical variation
of ~4 mm amplitude, from a true time series that
had no seasonal variation. Amplitudes were about
half as large for the site closer to the center of the
network.
Fig. 6 Apparent seasonal variations in a test site, which
had no seasonal variation in the true time series. The
apparent variations are due entirely to aliasing of seasonal variations in the sites used to realize the reference
frame, when only a linear model of site motion is used.
There is considerable complexity and time-variability in
this systematic error, which could be interpreted erroneously as time-varying tectonic signal or geophysical
loading.
The induced bias is roughly opposite in phase to
the seasonal variations in the frame sites: if the
frame sites are high in the middle of the year, the
test site is low. The aliased seasonal signal at the
test sites did not exactly match the seasonal at any
single frame site, but presumably reflects a
weighted average of the frame sites. This result
also suggests that if one of the frame sites is missing from the solution for a time, a subtle change in
the reference frame realization bias will result,
which might be interpreted as time-varying tectonic signal.
6 Discussion
The biases discussed in this paper do not result
from errors in the ITRF model, but instead from
the user’s realization of the frame. Where significant seasonal variations are present, the ITRF may
be considered limited in that it describes only the
5
average secular motion of sites, and not the seasonal variation. I have shown that seasonal variations have a significant impact on practical reference frame realization and thus on the use of the
frame. We should consider moving toward an
ITRF model that includes seasonal as well as secular terms to describe the motion of sites.
This study needs to be repeated on a global basis, because the impact of seasonal variations will
be different for a global network than for a continental-scale network. I hypothesize that the dominant effect on a global network will be the longwavelength seasonal variations that are now often
modeled through low degree harmonic loading
(Blewitt et al., 2001). With a continental-scale
network, the biases in time series from aliasing of
seasonal deformation may be largest in a case
where the seasonal deformation varies systematically across the network, for example due to snow
loading in the north, or where there is extreme
hydrological loading in one part of the network,
such as the Amazon Basin in South America.
These cases may induce erroneous “tilts” in the
user’s realization of the reference frame, and will
produce fictitious apparent seasonal variations in
other sites used in the same solution.
When a small network is used, and daily reference frame transformation parameters are estimated to align the solution with a linear model of
site motion, the result is a regional filter (e.g.,
Wdowinski et al., 1997). Regional filters can be
very effective in removing truly common-mode
seasonal and other variations, but if these variations are not truly common mode, then use of a
regional filter may produce a time series in which
the variations appear more temporally complex
than they actually are. Imagine a case where sites
in a network have similar seasonal variations but
there is a systematic change of the phase of the
variation across the network. Removing the average common mode variation with a regional filter
would introduce beat frequency variations in time
caused by differencing two similar but phase
shifted periodic signals. Removing seasonal variations using a realistic geophysical model would be
a better approach than removing most of the variations using a regional filter.
The impact of seasonal variations can be important even in cases where a user may not always
recognize that they are making a regional reference frame realization. For example, one common
practice is to estimate a velocity solution from a
time series of minimally constrained solutions.
This velocity solution would not be biased in itself, but if the time series of solutions was then
aligned with the velocity solution, the aligned time
Submitted to GRF2006 Symposium Proceedings, Springer Verlag IAG Series, 12/2006, Revised 1/2008
series may be biased if there are significant seasonal variations in the sites used to align the time
series.
7 Conclusions
Based on analysis of 6 years of a regional GPS
solution, seasonal variations in most GPS sites
equal or exceed 4-5 mm amplitude in height. The
seasonal variations at most sites are not sinusoids,
nor do the curves match an annual plus semiannual model well. At most sites, the negative part
of the seasonal variation in height is larger in amplitude than the positive part, but shorter in duration. In northern North America, the seasonal
variations are generally consistent with the expected phase from snow loading, and the amplitudes are reasonable based on other studies of
snow and hydrologic loading.
When a set of sites are used to estimate a transformation to align a solution with ITRF or a regional velocity solution, ignoring the seasonal
variations bias the reference frame parameters, and
thus the coordinates of all sites in the transformed
solution. A synthetic test shows that these biases
can have amplitudes as large as a few mm, and can
display considerable complexity in time.
When seasonal variations are present, reference
frame realization or alignment of time series to a
velocity solution should incorporate accurate seasonal variations for all sites used to estimate the
frame transformation. These variations should be
estimated on a global basis, and we should move
toward a time-dependent ITRF model that incorporates seasonal variation in site coordinates as
well as position and secular velocity.
Incorporating seasonal variations into the ITRF
model would not be simple. The first problem is
that some of the seasonal variation is due to technique-dependent systematic error rather than a
geophysical signal (Ray 2006). However, based on
the results shown here, I propose that the variations at sites that show large seasonal variations
are dominated by signal rather than techniquedependent noise. Further research is required to
test this hypothesis and isolate geophysical seasonal variations from systematic errors and noise.
Seasonal residuals to the ITRF model at co-located
multi-technique sites should be examined to test
whether they are coherent, which should give insight into the magnitude of noise and techniquedependent systematic errors.
I propose that empirical studies of seasonal
variation such as this study be combined with geophysical modeling to develop a set of first order
seasonal corrections to the linear ITRF model. I
6
suggest that deformation caused by geophysical
loading from load variations with dominant periods greater than tidal periods (greater than daily)
be incorporated into a seasonal variation model
rather than being incorporated into observation
models. That is, seasonal variation models should
include hydrologic loading, non-tidal ocean loading, and atmospheric loading. Geophysical models
for some of these components may be good
enough to provide a priori estimates of these deformations. Broad agreement on both the strategy
and particular models would be required before
seasonal variations can be incorporated into ITRF.
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